Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $a = \dfrac{5k^2 - 30k}{k + 5} \div \dfrac{k^2 - 16k + 60}{-3k + 30} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{5k^2 - 30k}{k + 5} \times \dfrac{-3k + 30}{k^2 - 16k + 60} $ First factor the quadratic. $a = \dfrac{5k^2 - 30k}{k + 5} \times \dfrac{-3k + 30}{(k - 6)(k - 10)} $ Then factor out any other terms. $a = \dfrac{5k(k - 6)}{k + 5} \times \dfrac{-3(k - 10)}{(k - 6)(k - 10)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ 5k(k - 6) \times -3(k - 10) } { (k + 5) \times (k - 6)(k - 10) } $ $a = \dfrac{ -15k(k - 6)(k - 10)}{ (k + 5)(k - 6)(k - 10)} $ Notice that $(k - 10)$ and $(k - 6)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ -15k\cancel{(k - 6)}(k - 10)}{ (k + 5)\cancel{(k - 6)}(k - 10)} $ We are dividing by $k - 6$ , so $k - 6 \neq 0$ Therefore, $k \neq 6$ $a = \dfrac{ -15k\cancel{(k - 6)}\cancel{(k - 10)}}{ (k + 5)\cancel{(k - 6)}\cancel{(k - 10)}} $ We are dividing by $k - 10$ , so $k - 10 \neq 0$ Therefore, $k \neq 10$ $a = \dfrac{-15k}{k + 5} ; \space k \neq 6 ; \space k \neq 10 $